\(f\in L^{s}(\mathbb {R}^{n}),\)  \(1\le s < \frac{n}{\alpha }\) . A well known result of Sobolev gives the
boundedness of \(I_{\alpha }\)  from \(L^{p}(\mathbb {R}^{n})\)  into \(L^{q}(\mathbb {R}^{n})\)  for \(1<p<\frac{n}{\alpha }\)  and \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}.\)  In [1]} C. Capone, D. Cruz Uribe and A. Fiorenza
extend this result to the case of Lebesgue spaces with variable exponents \(L^{p(.)}.\)  In [2]} E. Stein and G. Weiss used the theory of harmonic
functions of several variables to prove that these operators are bounded
from \(H^{1}(\mathbb {R}^{n})\)  into \(L^{\frac{n}{n-\alpha }}(\mathbb {R}^{n}).\) 
In [3]}, M. Taibleson and G. Weiss obtained the boundedness of the
Riesz potential \(I_{\alpha }\)  from the Hardy spaces \(H^{p}(\mathbb {R}^{n})\) 
into \(H^{q}(\mathbb {R}^{n}),\)  for \(0<p<1\)  and \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha }{n}.\)  We extend these results to the context of Hardy spaces with
variable exponents. The main tools that we use are Lemma 7 and the molecular
decomposition developed in [4]}.
